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dx 1₁ (x² + 8x + 13)² press numbers in exact form. Use symbolic notation and fracti - C as much as possible.)

dx 1₁ (x² + 8x + 13)² press numbers in exact form. Use symbolic notation and fracti - C as much as possible.)

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter7: Integration
Section7.3: Area And The Definite Integral
Problem 1E: Explain the difference between an indefinite integral and a definite integral.
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Evaluate the integral by completing the square as a first step.

 

∫??(?2+8?+13)2∫dx(x2+8x+13)2

 

(Express numbers in exact form. Use symbolic notation and fractions where needed. Use ?C for the arbitrary constant. Absorb into ?C as much as possible.)

--- ### Completing the Square Method in Integration Evaluate the integral by completing the square as a first step: \[ \int \frac{dx}{(x^2 + 8x + 13)^2} \] (Express numbers in exact form. Use symbolic notation and fractions where needed. Use \( C \) for the arbitrary constant. Absorb into \( C \) as much as possible.) --- The original attempt to solve the integral is displayed below: \[ \int \frac{dx}{(x^2 + 8x + 13)^2} = \frac{1}{12\sqrt{3}} \ln \left| x + 4 + \sqrt{3} \right| - \frac{1}{12} \frac{1}{(x + 4 + \sqrt{3})} - \frac{1}{12\sqrt{3}} \ln \left| x + 4 - \sqrt{3} \right| - \frac{1}{12} \frac{1}{(x + 4 - \sqrt{3})} + C \] (Note: The result is marked as incorrect). --- ### Explanation for Completing the Square To evaluate the integral correctly, the method of completing the square is utilized to transform the quadratic expression in the denominator into a more integrable form. 1. **Complete the square for \(x^2 + 8x + 13\)**: \[ x^2 + 8x + 13 = (x^2 + 8x + 16) - 3 = (x + 4)^2 - 3 \] 2. **Rewrite the integral**: \[ \int \frac{dx}{((x + 4)^2 - 3)^2} \] ### Continuing the Integration This process breaks down the more challenging quadratic expression, allowing for easier integration using appropriate substitution methods or known integral forms. --- For more detailed step-by-step solutions and explanations, please refer to the instructional videos and practice problems provided on our website. ---
Transcribed Image Text:--- ### Completing the Square Method in Integration Evaluate the integral by completing the square as a first step: \[ \int \frac{dx}{(x^2 + 8x + 13)^2} \] (Express numbers in exact form. Use symbolic notation and fractions where needed. Use \( C \) for the arbitrary constant. Absorb into \( C \) as much as possible.) --- The original attempt to solve the integral is displayed below: \[ \int \frac{dx}{(x^2 + 8x + 13)^2} = \frac{1}{12\sqrt{3}} \ln \left| x + 4 + \sqrt{3} \right| - \frac{1}{12} \frac{1}{(x + 4 + \sqrt{3})} - \frac{1}{12\sqrt{3}} \ln \left| x + 4 - \sqrt{3} \right| - \frac{1}{12} \frac{1}{(x + 4 - \sqrt{3})} + C \] (Note: The result is marked as incorrect). --- ### Explanation for Completing the Square To evaluate the integral correctly, the method of completing the square is utilized to transform the quadratic expression in the denominator into a more integrable form. 1. **Complete the square for \(x^2 + 8x + 13\)**: \[ x^2 + 8x + 13 = (x^2 + 8x + 16) - 3 = (x + 4)^2 - 3 \] 2. **Rewrite the integral**: \[ \int \frac{dx}{((x + 4)^2 - 3)^2} \] ### Continuing the Integration This process breaks down the more challenging quadratic expression, allowing for easier integration using appropriate substitution methods or known integral forms. --- For more detailed step-by-step solutions and explanations, please refer to the instructional videos and practice problems provided on our website. ---
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